Thursday, March 10, 2011

The latest updates

To The Continuum (roughly, the real number line):
Reuben Hersch, “Some Proposals…” (1979); repr. in Thomas Tymoczko, ed., New Directions in the Philosophy of Mathematics (1986, rev. 1998), p. 13:
If we teach our students anything at all about the philosophical problems of mathematics, it is that there is only one problem of interest (the problem of the foundation of the real number system), and that problem seems totally intractable.

~ ~ ~

Analytical appendix:

‘Twere a mug’s game, to cite specific instances of sociobiological overreaching -- just-so stories that purport to explain Love, Music, Art, what have you.  Chesterton already skewered these  several generations back.   More worth noting are the (rare) cases where such thumb-sucking is found among mathematicians themselves.
Thus Reuben Hersch (apparently during a brief psychotic episode) wrote (“Some Proposals…” (1979); repr. in Thomas Tymoczko, ed., New Directions in the Philosophy of Mathematics (1986, rev. 1998), p. 23):

Our mathematical ideas fit the world  for the same reason that our lungs are suited to the atmosphere of this planet.

Yet the author knows better.  Just a bit further up the page (with his customary lucidity) he wrote:

Consider the theorem  2^c < 2^(2^c), or any theorem in homological algebra.  No philosopher has yet explained in what sense such theorems should be regarded as referring to physical ‘possibilities’.

They are thus devoid of selective advantage.
(That other, simpler, arithmetical abilities may indeed have survival value, is demonstrated  beyond all rebuttal  here.)

Hersh’s initial statement is rather like saying that the observed patterns of conic-section orbits fit the gravitational inverse-square law as result of evolution:  these do indeed cohere, but not like that.  To that he adds a kind of category mistake, replacing  “the actual mathematical facts” with “our mathematical ideas”.   The math (in certain of its aspects) matches the world, you might say, and always has.   Our ideas have over time come slowly to discover and partially appreciate a handful of these mathematical patterns, rather the way we have come, in time, to discover the New World, or quasars, or quarks.   Just how we can do this, and why we wish to -- neglecting, at times, our own physical well-being, and the hope of progeny, for the sake of the purely ethereal and extra-worldly attractions of homological algebra -- is a nice conundrum, to be discussed over brandy.  I rather suspect -- ah, but we don’t want to spoil it.

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